HSSLIVE Plus One Maths Chapter 10: Straight Lines Notes

The study of straight lines connects algebra and geometry, providing a foundation for analytic geometry. This chapter covers various forms of line equations, angle between lines, distance formulas, and applications in modeling linear relationships. Students will master techniques for finding equations of lines under different conditions and solve geometric problems using algebraic methods. These concepts are fundamental to understanding more complex curves, transformations in the plane, and serve as building blocks for multivariable calculus and linear algebra.

Chapter 10: Straight Lines

Coordinate Geometry Basics:

• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
• Section Formula:
  • Internal division: (x, y) = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)]
  • External division: (x, y) = [(mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n)]
• Mid-point Formula: (x, y) = [(x₁+x₂)/2, (y₁+y₂)/2]

Slope of a Line:

• Definition: m = tan θ = (y₂-y₁)/(x₂-x₁)
• Properties:
  • Parallel lines have equal slopes: m₁ = m₂
  • Perpendicular lines: m₁ × m₂ = -1

Forms of Line Equations:

1. Slope-Intercept Form: y = mx + c
   • m is the slope
   • c is the y-intercept
2. Point-Slope Form: y – y₁ = m(x – x₁)
   • (x₁, y₁) is a point on the line
   • m is the slope
3. Two-Point Form: (y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁)
   • (x₁, y₁) and (x₂, y₂) are points on the line
4. Intercept Form: x/a + y/b = 1
   • a is the x-intercept
   • b is the y-intercept
5. General Form: ax + by + c = 0

Distance from a Point to a Line:

Distance from point (x₁, y₁) to line ax + by + c = 0 is: d = |ax₁ + by₁ + c|/√(a² + b²)

Angle Between Two Lines:

If m₁ and m₂ are slopes of two lines, then: tan θ = |(m₂ – m₁)/(1 + m₁m₂)|